{
  "abstract": "A majority C-coloring of a graph $G$ assigns colors to the vertices such that every vertex shares its color with at least half of its neighbors. The maximum number of colors that can be used in such a coloring of $G$ is denoted by $\\overlineχ_{\\geqslant}(G)$. In this paper, the focus is on the majority C-coloring in Cartesian product graphs. It is shown that $\\overlineχ_{\\geqslant}(G \\square H) \\ge \\overlineχ_{\\geqslant}(G) \\overlineχ_{\\geqslant}(H)$ gives a sharp lower bound, but the difference also can be arbitrarily large. For two-dimensional Hamming graphs, the exact value $\\overlineχ_{\\geqslant}(K_m \\square K_n) = \\min\\{m,n\\}$ is established. Balanced Hamming graphs of higher dimension, that is the $k$th powers of complete graphs with respect to the Cartesian product, are also studied. It is proved that $\\overlineχ_{\\geqslant}(K_n^{\\square, k})= n^{k/2}$ holds for every even integer $k$. If $k$ is odd and the Hamming graph is the $k$-dimensional hypercube, then $\\overlineχ_{\\geqslant}(K_2^{\\square, k})= 2^{\\lfloor k/2\\rfloor}$. On the other hand, a majority C-coloring of $K_n^{\\square, k}$ with at least $3 n^{\\lfloor k/2\\rfloor}/2 $ colors is presented for every $n \\ge 7$ and odd $k \\ge 3$. For Cartesian grids, the main result shows that $\\overlineχ_{\\geqslant}(P_m \\square P_n) = 1 + \\lfloor m/2\\rfloor \\lfloor n/2\\rfloor$ if at least one of $m$ and $n$ is odd, while $\\overlineχ_{\\geqslant}(P_m \\square P_n)=mn/4$ holds if both parameters are even and $m \\ge n \\ge 4$. The paper concludes with a conjecture and several open problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.27669",
  "author_total": 5,
  "authors": [
    {
      "name": "Csilla Bujtás"
    },
    {
      "name": "Magda Dettlaff"
    },
    {
      "name": "Hanna Furmańczyk"
    },
    {
      "name": "Aleksandra Laskowska"
    },
    {
      "name": "Zsolt Tuza"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dty12xmds6trv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.27669",
  "fulltext_url": null,
  "id": "dty12xmds6trv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.27669",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7204764386",
  "pdf_url": "https://arxiv.org/pdf/2608.27669",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dty12xmds6trv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2608.27669",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.27669"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Majority C-coloring in Cartesian products",
  "title_text": "Majority C-coloring in Cartesian products",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/dty12xmds6trv",
  "venue": "arXiv",
  "versions": []
}